FIG. 1 diagrammatically illustrates pertinent portions of an exemplary digital communication system according to the prior art. In the example of FIG. 1, a digital transmitter produces a sequence of iid (independent identical distributed) symbols designated as A[n]. This sequence of symbols is transmitted in a digital communication burst across an equivalent channel to an equalizer 11 provided in a digital receiver. As illustrated in FIG. 1, the equivalent channel between the transmitted symbols A[n] and the equalizer input includes portions of both the digital transmitter and the digital receiver, as well as a physical communication link or channel. For example, the equivalent channel would include modulation components of the transmitter, and demodulation components and notch filters of the receiver.
The equalizer 11 includes a feed forward equalizer (FFE) 13, which is a finite impulse response (FIR) filter that processes the received samples after some pre-processing in earlier stages of the receiver (included in the equivalent channel of FIG. 1). The sampling rate of the input to this feed forward equalizer module can be one sample per symbol or more. A sampling rate of one sample per symbol matches exactly the symbol rate (also known as the baud rate), and a feed forward equalizer having a one sample per symbol sampling rate is conventionally referred to as a T-spaced equalizer (TSE). A feed forward equalizer whose input sampling rate is higher than the symbol rate (for example twice the symbol rate) is conventionally referred to as a fractionally spaced equalizer (FSE). An upsampler 18 illustrates the possibility of multiplying the sampling rate by M at the transmitter, in which case a corresponding downsampler 14 divides the sampling rate by M at the receiver.
A slicer 15 is the decision element of the equalizer. The output of the feed forward equalizer module 13 is combined with information from a feedback loop 12 to produce the input for the slicer 15. The output Z[n] of the slicer 15 is indicative of which symbol (in the constellation) was transmitted. One typical conventional decision criteria that can be used by the slicer 15 is the Minimum Euclidean Distance (MED) criteria. The output of the slicer 15 is also provided as an input to the feedback loop 12.
The feedback loop 12 includes a feedback filter 17, for example a FIR filter that operates at the symbol rate. The input to this feedback filter 17 is the slicer output, namely the decision regarding what symbol was last transmitted. Assuming correct decisions from the slicer 15, the feedback filter 17 can compensate for the post cursor part of the interference from symbols that are previous in time. Such interference from timewise previous symbols is conventionally known as inter-symbol interference or ISI.
The FIR filter of the feed forward equalizer 13 is defined by a finite length sequence C[n] of filter coefficients (or filter taps), and the FIR filter of the feedback filter module 17 is defined by a finite length sequence D[n] of filter coefficients. The length of C[n] is herein referred to as K, and the length of D[n] is herein referred to as L. The basic task of the equalizer is to find the coefficients of C[n] and D[n] that minimize the mean squared error (MSE) associated with the decided symbols in the slicer output Z[n]. A coefficient determiner 19 uses conventional techniques, for example, least mean squares (LMS), least squares (LS) or recursive least squares (RLS) to determine the coefficients of C[n] and D[n] that minimize the MSE of the slicer 15.
FIG. 2 diagrammatically illustrates a conventional model of the equivalent channel of FIG. 1. The equivalent channel is modeled as a digital filter 21 having a filter tap sequence H[n], and an additive noise factor. Assume for purposes of exemplary exposition that the additive noise in the equivalent channel model is stationary white noise w[n] that obeys the following statistics:
Equation 1:E{w[n]}=0
Equation 2:       Rw    ⁡          [      l      ]        =            E      ⁢              {                              w            ⁡                          [              n              ]                                ·                      w            ⁡                          [                              n                -                1                            ]                                      }              =          {                           ⁢                                                  N0              /              2                                                          l              =              0                                                            0                                otherwise                              ⁢                           }      where E is the expected value operator, N0/2 is the noise variance, R is the autocorrelation function, and l is the time index.
The source A[n] of FIG. 1 is also white (i.e., independent identical distributed) and obeys the following statistics:Equation 3:       E    ⁢          {                        A          n                ·                  A                      n            -            m                              }        =      {                   ⁢                                        σ            A            2                                                m            =            0                                                0                          otherwise                      ⁢                   }  where ρA2 is the signal variance.
Under the exemplary conditions described above, the MSE of the slicer output can be expressed as follows for the case of a T-spaced equalizer:Equation 4:   MSE  =            E      ⁢              {                              (                                          Z                ⁡                                  [                  n                  ]                                            -                              A                ⁡                                  [                  n                  ]                                                      )                    2                }              =                            σ          A          2                ·                              ∑                          m              =                              -                ∞                                                    -              1                                ⁢                                           ⁢                                    (                                                ∑                                      k                    =                                          -                      U                                                        V                                ⁢                                                                   ⁢                                                      C                    ⁡                                          [                      k                      ]                                                        ·                                      h                    ⁡                                          [                                              m                        -                        k                                            ]                                                                                  )                        2                              +                                    σ            A            2                    ⁡                      (                                          (                                                      ∑                                          k                      =                                              -                        U                                                              V                                    ⁢                                                                           ⁢                                                            C                      ⁡                                              [                        k                        ]                                                              ·                                          h                      ⁡                                              [                                                  -                          k                                                ]                                                                                            )                            -              1                        )                          2            +                        σ          A          2                ·                              ∑                          m              =              1                        ∞                    ⁢                                           ⁢                                    (                                                                    ∑                                          k                      =                                              -                        U                                                              V                                    ⁢                                                                           ⁢                                                            C                      ⁡                                              [                        k                        ]                                                              ·                                          h                      ⁡                                              [                                                  m                          -                          k                                                ]                                                                                            +                                  D                  ⁡                                      [                    m                    ]                                                              )                        2                              +                                    N            ⁢                                                   ⁢            0                    2                ·                              ∑                          k              =                              -                U                                      V                    ⁢                                           ⁢                                    C              ⁡                              [                k                ]                                      2                              
For the fractionally spaced equalizer, the MSE can be expressed as follows:Equation 5:   MSE  =            E      ⁢              {                              (                                          Z                ⁡                                  [                  n                  ]                                            -                              A                ⁡                                  [                  n                  ]                                                      )                    2                }              =                            σ          A          2                ·                              ∑                          m              =                              -                ∞                                                    -              1                                ⁢                                           ⁢                                    (                                                ∑                                      k                    =                                          -                      U                                                        V                                ⁢                                                                   ⁢                                                      C                    ⁡                                          [                      k                      ]                                                        ·                                      h                    ⁡                                          [                                                                        2                          ⁢                          m                                                -                        k                                            ]                                                                                  )                        2                              +                                    σ            A            2                    ⁡                      (                                          (                                                      ∑                                          k                      =                                              -                        U                                                              V                                    ⁢                                                                           ⁢                                                            C                      ⁡                                              [                        k                        ]                                                              ·                                          h                      ⁡                                              [                                                  -                          k                                                ]                                                                                            )                            -              1                        )                          2            +                        σ          A          2                ·                              ∑                          m              =              1                        ∞                    ⁢                                           ⁢                                    (                                                                    ∑                                          k                      =                                              -                        U                                                              V                                    ⁢                                                                           ⁢                                                            C                      ⁡                                              [                        k                        ]                                                              ·                                          h                      ⁡                                              [                                                                              2                            ⁢                            m                                                    -                          k                                                ]                                                                                            +                                  D                  ⁡                                      [                    m                    ]                                                              )                        2                              +                                    N            ⁢                                                   ⁢            0                    2                ·                              ∑                          k              =                              -                U                                      V                    ⁢                                           ⁢                                    C              ⁡                              [                k                ]                                      2                              
For a T-spaced equalizer, Equation 4 is the cost function that is to be minimized with respect to C[n] and D[n]. In the case of a fractionally spaced equalizer, Equation 5 is to be minimized with respect to C[n] and D[n].
In Equations 4 and 5 and hereinafter, U is the number of uncausal coefficients in C[n], and V is the number of causal coefficients in C[n].
For purposes of exemplary exposition, the following conditions are assumed. Communications occur in bursts through the equivalent channel of FIG. 1, which equivalent channel can vary between two consecutive bursts. A single burst includes a preamble (a short, unique training sequence) and an unknown sequence of symbols (the substantive information), and the channel can be easily estimated from the preamble at the beginning of the burst. It is desirable to decode the burst with minimum latency, and the number of MMACs (Million Multiply and Accumulate per second) is of course limited. Under these conditions, the receiver has a limited time and a limited number of data processing operations to achieve a sufficiently low MSE.
Conventional techniques which can be implemented by the coefficient determiner of FIG. 1 to determine the C[n] and D[n] coefficients which will minimize the MSE include the least mean square (LMS) algorithm and its various versions. To use these decision directed algorithms, the MSE between the slicer input and the slicer output must be lower than a certain threshold (which threshold depends on the constellation being used) in order to permit the decoding of the unknown symbols to commence with an acceptably low symbol error rate. Most LMS algorithms therefore have the disadvantage of potentially requiring many iterations and many repetitions over the preamble to achieve the desired results.
The conventional LS or RLS algorithm can be used to determine the C[n] and D[n] coefficients, but this algorithm is more complex and therefore less practical than the LMS algorithm. If the channel is unknown, implementation of the LS/RLS approach involves matrix inversion of the size K+L, where K is the length of the sequence C[n] and L is the length of the sequence D[n]. If the channel is known, then a matrix inversion of size K is required. Although the LS/RLS, algorithm provides the optimal solution (based on the empirical statistics of the preamble) in terms of MSE, the aforementioned matrix inversions are quite expensive in terms of the required data processing power (e.g. multiply and accumulate operations).
It is desirable in view of the foregoing to provide for equalization that requires less computational complexity than the prior art, while still maintaining sufficient performance.
The present invention provides efficient equalization by initializing the values of only a subset of the feed forward and feedback coefficients. After these values are initialized, decoding of the unknown symbols is commenced using, for example, the LMS algorithm or another iterative, decision directed algorithm to determine all of the feed forward and feedback coefficients. The initialization of a subset of the coefficient values before decoding the unknown symbols can advantageously provide an acceptable trade-off between computational complexity and equalization performance.